Compute Taylor Series for Univariate Expressions

Note:
Use only in the MuPAD Notebook Interface. This
functionality does not run in MATLAB.

Taylor series expansions serve for approximating an arbitrary
expression by a polynomial expression around some value of a variable.
Taylor series expansions approximate expressions for which the derivatives
up to infinite order exist around a particular value x0 of
a variable x:

.

To compute Taylor series expansion, use the taylor command. For example, approximate
the expression sin(x)/x around x = 0:

exact := sin(x)/x:
approx := taylor(sin(x)/x, x)

Plot the exact expression and its taylor series expansion in
the same coordinate system. The taylor series expansion approximates
the expression near x = 0, but visibly deviates
from sin(x)/x for larger |x|:

Taylor series expansions around x = 0 are
also called Maclaurin series expansions. Approximate the expressions
by Maclaurin series:

taylor(exp(x), x);
taylor(sin(x), x);
taylor(cos(x)/(1 - x), x)

The Maclaurin series expansion does not exist for the following
expression. MuPAD® throws an error:

taylor(arccot(x), x)

Error: Cannot compute a Taylor expansion of 'arccot(x)'. Try 'series' for a more general expansion. [taylor]

You can represent the following expression by a Taylor series
around x = 1. To compute the series expansion around
a nonzero value of a variable, specify the value. For example, compute
the Taylor series expansions around x = 1 for the
following expressions:

taylor(ln(x), x = 1);
taylor(arccot(x), x = 1)

The taylor command
returns results in the form of Taylor series including the order term O. To convert
the results to a regular polynomial expression without the O-term, use
the expr command: